Problem: A secant line intersects the graph of $f(x)=2^x$ at two points with $x$ -coordinates $3$ and $t$. What is the slope of the secant line? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{2^{t-3}}{t-3}$ (Choice B) B $\dfrac{2^t-2^3}{3}$ (Choice C) C $\dfrac{2^t-2^3}{t-3}$ (Choice D) D $\dfrac{2^{t-3}}{t}$
Solution: We are given that the secant line intersects the graph at $x=3$ and $x=t$. Since these points are on the graph of $f(x)=2^x$, we know that they must be $(3, 2^{{3}})$ and $(t, 2^{{t}})$, respectively. This should be enough to find the slope of that line. $\begin{aligned} \text{Slope}&=\dfrac{\text{Change in }y}{\text{Change in }x} \\\\ &=\dfrac{2^t-2^3}{t-3} \end{aligned}$ In conclusion, the slope of the secant line is $\dfrac{2^t-2^3}{t-3}$.